How many musical scales are there?

How many musical scales are there? That’s not a simple question. It depends on how you define “scale.”

For this post, I’ll only consider scales starting on C. That is, I’ll only consider changing the intervals between notes, not changing the starting note. Also, I’ll only consider subsets of the common chromatic scale; this post won’t get into dividing the octave into more or less than 12 intervals.

First of all we have the major scale — C D E F G A B C — and the “natural” minor scale: A B C D E F G A. The word “natural” suggests there are other minor scales. More on that later.

Then we have the classical modes: Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian. These have the same intervals as taking the notes of the C major scale and starting on D, E, F, G, A, and B respectively. For example, Dorian has the same intervals as D E F G A B C D. Since we said we’d start everything on C, the Dorian mode would be C D E♭ F G A B♭ C. The Aeloian mode is the same as the natural minor scale.

The harmonic minor scale adds a new wrinkle: C D E♭ F G A♭ B C. Notice that A♭ and B are three half steps apart. In all the scales above, notes were either a half step or a whole step apart. Do we want to consider scales that have such large intervals? It seems desirable to include the harmonic minor scale. But what about this: C E♭ G♭ A C. Is that a scale? Most musicians would think of that as a chord or arpeggio rather than a scale. (It’s a diminished seventh chord. And it would be more common to write the A as a B♭♭.)

We might try to put some restriction on the definition of a scale so that the harmonic minor scale is included and the diminished seventh arpeggio is excluded. Here’s what I settled on. For the purposes of this post, I’ll say that a scale is an ascending sequence of eight notes with two restrictions: the first and last are an octave apart, and no two consecutive notes are more than three half steps apart. This will include modes mentioned above, and the harmonic minor scale, but will exclude the diminished seventh arpeggio. (It also excludes pentatonic scales, which we may or may not want to include.)

One way to enumerate all possible scales would be to start with the chromatic scale and decide which notes to keep. Write out the notes C, C♯, D, … , B, C and write a ‘1’ underneath a note if you want to keep it and a ‘0’ otherwise. We have to start and end on C, so we only need to specify which of the 11 notes in the middle we are keeping. That means we can describe any potential scale as an 11-bit binary number. That’s what I did to carry out an exhaustive search for scales with a little program.

There are 266 scales that meet the criteria listed here. I’ve listed all of them on another page. Some of these scales have names and some don’t. I’ve noted some names as shown below. I imagine there are others that have names that I have not labeled. I’d appreciate your help filling these in.

|--------------+-----------------------+-------------------|
| Scale number | Notes                 | Name              |
|--------------+-----------------------+-------------------|
|          693 | C D  E  F# G  A  B  C | Lydian mode       |
|          725 | C D  E  F  G  A  B  C | Major             |
|          726 | C D  E  F  G  A  Bb C | Mixolydian mode   |
|          825 | C D  Eb F# G  Ab B  C | Hungarian minor   |
|          826 | C D  Eb F# G  Ab Bb C | Ukrainian Dorian  |
|          854 | C D  Eb F  G  A  Bb C | Dorian mode       |
|          858 | C D  Eb F  G  Ab Bb C | Natural minor     |
|         1235 | C Db E  F  G  Ab B  C | Double harmonic   |
|         1242 | C Db E  F  G  Ab Bb C | Phrygian dominant |
|         1257 | C Db E  F  Gb Ab B  C | Persian           |
|         1370 | C Db Eb F  G  Ab Bb C | Phrygian mode     |
|         1386 | C Db Eb F  Gb Ab Bb C | Locrian mode      |
|--------------+-----------------------+-------------------|

Update: See this page for Pete White’s free ebook listing all possible scales with 3 to 11 notes.

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